Final answer:
To determine the speed of fluid leaving the opening at the bottom of the tank, we can use Bernoulli's equation. The equation shows that the speed of the fluid leaving the opening is independent of the size of the opening and is equal to the speed it would have if it fell a distance h from the surface of the reservoir. This applies when there is negligible resistance.
Step-by-step explanation:
To determine the speed of fluid leaving the opening at the bottom of the tank, we can use Bernoulli's equation. Bernoulli's equation states that the sum of the pressure energy, kinetic energy, and potential energy of a fluid remains constant along a streamline. In this case, we consider the water flowing from the surface of the reservoir to the outlet of the tube. Therefore, we can write the equation as:
P₁ + 1/2ρv₁² + ρgh₁ = P₂ + 1/2ρv₂² + ρgh₂
In this equation, P₁ and P₂ represent the pressures at points 1 and 2, ρ is the density of the fluid, v₁ and v₂ are the speeds of the fluid at points 1 and 2, and h₁ and h₂ are the heights of the fluid at points 1 and 2. We can assume that the pressure at the surface of the reservoir is atmospheric pressure, and at the outlet of the tube, the pressure is also atmospheric. Therefore, we can cancel out the pressure terms, and the equation becomes:
1/2ρv₁² + ρgh₁ = 1/2ρv₂² + ρgh₂
Since the fluid is incompressible, we can assume that the density remains constant throughout the system. Additionally, since the flow is nearly steady, we can assume that the height difference between points 1 and 2 is negligible. Therefore, we can simplify the equation as:
1/2v₁² = 1/2v₂²
This equation shows that the speed of the fluid leaving the opening at the bottom is independent of the size of the opening, and is equal to the speed it would have if it fell a distance h from the surface of the reservoir, as long as the resistance is negligible.